Embedded newform 96.3.p.a.5.5

• Label: 96.3.p.a.5.5
• Level: $96$
• Weight: $3$
• Character: 96.5
• Analytic conductor: $2.616$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	96.p (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.61581053786\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			5.5
Character	\(\chi\)	\(=\)	96.5
Dual form			96.3.p.a.77.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.76181 + 0.946587i) q^{2} +(-2.17853 + 2.06252i) q^{3} +(2.20795 - 3.33541i) q^{4} +(2.46183 - 5.94339i) q^{5} +(1.88580 - 5.69594i) q^{6} +(0.814968 + 0.814968i) q^{7} +(-0.732725 + 7.96637i) q^{8} +(0.491998 - 8.98654i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\)	\(31\)	\(37\)	\(65\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{8}\right)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.76181	+	0.946587i	−0.880905	+	0.473293i
							\(3\)	−2.17853	+	2.06252i	−0.726177	+	0.687508i
\(5\)	2.46183	−	5.94339i	0.492366	−	1.18868i								−0.461146	−	0.887324i	\(-0.652562\pi\)
							0.953513	−	0.301353i	\(-0.0974383\pi\)
\(7\)	0.814968	+	0.814968i	0.116424	+	0.116424i	0.762919	−	0.646495i	\(-0.223766\pi\)
							−0.646495	+	0.762919i	\(0.723766\pi\)
\(11\)	19.5212	+	8.08595i	1.77466	+	0.735086i	0.993904	+	0.110245i	\(0.0351634\pi\)
							0.780751	+	0.624842i	\(0.214837\pi\)
\(13\)	−4.36879	+	1.80961i	−0.336061	+	0.139201i	−0.544331	−	0.838870i	\(-0.683217\pi\)
							0.208270	+	0.978071i	\(0.433217\pi\)
\(17\)	17.8870			1.05218			0.526088	−	0.850430i	\(-0.323658\pi\)
							0.526088	+	0.850430i	\(0.323658\pi\)
\(19\)	18.5864	−	7.69875i	0.978234	−	0.405198i	0.164463	−	0.986383i	\(-0.447411\pi\)
							0.813771	+	0.581186i	\(0.197411\pi\)
\(23\)	−24.7917	−	24.7917i	−1.07790	−	1.07790i	−0.996698	−	0.0812004i	\(-0.974125\pi\)
							−0.0812004	−	0.996698i	\(-0.525875\pi\)
\(29\)	19.7993	−	8.20114i	0.682734	−	0.282798i	−0.0142350	−	0.999899i	\(-0.504531\pi\)
							0.696969	+	0.717101i	\(0.254531\pi\)
\(31\)	6.09042			0.196465			0.0982325	−	0.995163i	\(-0.468681\pi\)
							0.0982325	+	0.995163i	\(0.468681\pi\)
\(37\)	3.01265	+	1.24788i	0.0814230	+	0.0337265i	0.423023	−	0.906119i	\(-0.360969\pi\)
							−0.341600	+	0.939845i	\(0.610969\pi\)
\(41\)	−22.8876	−	22.8876i	−0.558235	−	0.558235i	0.370570	−	0.928805i	\(-0.379162\pi\)
							−0.928805	+	0.370570i	\(0.879162\pi\)
\(43\)	−13.2465	+	31.9798i	−0.308058	+	0.743717i	0.691710	+	0.722175i	\(0.256858\pi\)
							−0.999768	+	0.0215418i	\(0.993142\pi\)
\(47\)	−45.8564			−0.975668			−0.487834	−	0.872936i	\(-0.662213\pi\)
							−0.487834	+	0.872936i	\(0.662213\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.3.p.a.5.5	✓	120
3.2	odd	2	inner	96.3.p.a.5.26	yes	120
4.3	odd	2		384.3.p.a.113.23		120
12.11	even	2		384.3.p.a.113.1		120
32.13	even	8	inner	96.3.p.a.77.26	yes	120
32.19	odd	8		384.3.p.a.17.1		120
96.77	odd	8	inner	96.3.p.a.77.5	yes	120
96.83	even	8		384.3.p.a.17.23		120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.p.a.5.5	✓	120	1.1	even	1	trivial
96.3.p.a.5.26	yes	120	3.2	odd	2	inner
96.3.p.a.77.5	yes	120	96.77	odd	8	inner
96.3.p.a.77.26	yes	120	32.13	even	8	inner
384.3.p.a.17.1		120	32.19	odd	8
384.3.p.a.17.23		120	96.83	even	8
384.3.p.a.113.1		120	12.11	even	2
384.3.p.a.113.23		120	4.3	odd	2